15 research outputs found
Stabilized plethysms for the classical Lie groups
The plethysms of the Weyl characters associated to a classical Lie group by
the symmetric functions stabilize in large rank. In the case of a power sum
plethysm, we prove that the coefficients of the decomposition of this
stabilized form on the basis of Weyl characters are branching coefficients
which can be determined by a simple algorithm. This generalizes in particular
some classical results by Littlewood on the power sum plethysms of Schur
functions. We also establish explicit formulas for the outer multiplicities
appearing in the decomposition of the tensor square of any irreducible finite
dimensional module into its symmetric and antisymmetric parts. These
multiplicities can notably be expressed in terms of the Littlewood-Richardson
coefficients
Parabolic Kazhdan-Lusztig polynomials, plethysm and gereralized Hall-Littlewood functions for classical types
We use power sums plethysm operators to introduce H functions which
interpolate between the Weyl characters and the Hall-Littlewood functions Q'
corresponding to classical Lie groups. The coefficients of these functions on
the basis of Weyl characters are parabolic Kazhdan-Lusztig polynomials and
thus, are nonnegative. We prove that they can be regarded as quantizations of
branching coefficients obtained by restriction to certain Levi subgroups. The H
functions associated to linear groups coincide with the polynomials introduced
by Lascoux Leclerc and Thibon (LLT polynomials).Comment: To appear in European Journal of Combinatoric
Geometric complexity theory and matrix powering
Valiant's famous determinant versus permanent problem is the flagship problem in algebraic complexity theory. Mulmuley and Sohoni (Siam J Comput 2001, 2008) introduced geometric complexity theory, an approach to study this and related problems via algebraic geometry and representation theory. Their approach works by multiplying the permanent polynomial with a high power of a linear form (a process called padding) and then comparing the orbit closures of the determinant and the padded permanent. This padding was recently used heavily to show no-go results for the method of shifted partial derivatives (Efremenko, Landsberg, Schenck, Weyman, 2016) and for geometric complexity theory (Ikenmeyer Panova, FOCS 2016 and B\"urgisser, Ikenmeyer Panova, FOCS 2016). Following a classical homogenization result of Nisan (STOC 1991) we replace the determinant in geometric complexity theory with the trace of a variable matrix power. This gives an equivalent but much cleaner homogeneous formulation of geometric complexity theory in which the padding is removed. This radically changes the representation theoretic questions involved to prove complexity lower bounds. We prove that in this homogeneous formulation there are no orbit occurrence obstructions that prove even superlinear lower bounds on the complexity of the permanent. This is the first no-go result in geometric complexity theory that rules out superlinear lower bounds in some model. Interestingly---in contrast to the determinant---the trace of a variable matrix power is not uniquely determined by its stabilizer
Multipartite Quantum States and their Marginals
Subsystems of composite quantum systems are described by reduced density
matrices, or quantum marginals. Important physical properties often do not
depend on the whole wave function but rather only on the marginals. Not every
collection of reduced density matrices can arise as the marginals of a quantum
state. Instead, there are profound compatibility conditions -- such as Pauli's
exclusion principle or the monogamy of quantum entanglement -- which
fundamentally influence the physics of many-body quantum systems and the
structure of quantum information. The aim of this thesis is a systematic and
rigorous study of the general relation between multipartite quantum states,
i.e., states of quantum systems that are composed of several subsystems, and
their marginals. In the first part, we focus on the one-body marginals of
multipartite quantum states; in the second part, we study general quantum
marginals from the perspective of entropy.Comment: PhD thesis, ETH Zurich. The first part contains material from
arXiv:1208.0365, arXiv:1204.0741, and arXiv:1204.4379. The second part is
based on arXiv:1302.6990 and arXiv:1210.046
Wreath Macdonald operators
We construct a novel family of difference-permutation operators and prove
that they are diagonalized by the wreath Macdonald -polynomials; the
eigenvalues are written in terms of elementary symmetric polynomials of
arbitrary degree. Our operators arise from integral formulas for the action of
the horizontal Heisenberg subalgebra in the vertex representation of the
corresponding quantum toroidal algebraComment: v1, 47pp. Comments are welcome
An Invitation to the Generalized Saturation Conjecture
We report about some results, interesting examples, problems and conjectures
revolving around the parabolic Kostant partition functions, the parabolic
Kostka polynomials and ``saturation'' properties of several generalizations of
the Littlewood--Richardson numbers.Comment: 79 pages, new sections, new results and example